scholarly journals A conjectural refinement of strong multiplicity one for GL(n)

2021 ◽  
pp. 1-16
Author(s):  
Nahid Walji
Keyword(s):  
Number Field ◽  
Lower Bounds ◽  
Automorphic Representations ◽  
Hecke Eigenvalues ◽  
Multiplicity One ◽  
Strong Multiplicity ◽  
Cuspidal Automorphic Representations

Given a pair of distinct unitary cuspidal automorphic representations for GL([Formula: see text]) over a number field, let [Formula: see text] denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this paper, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin–Selberg products imply lower bounds on the size of [Formula: see text]. We also obtain further results for GL(3).

scholarly journals Unitary dual of GL(n) at archimedean places and global Jacquet–Langlands correspondence

2010 ◽  
Vol 146 (5) ◽  
pp. 1115-1164 ◽  
Author(s):  
A. I. Badulescu ◽  
D. Renard
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
General Linear Groups ◽  
Multiplicity One ◽  
Unitary Dual ◽  
Strong Multiplicity ◽  
Local Results

AbstractIn a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

scholarly journals The sign of Galois representations attached to automorphic forms for unitary groups

2011 ◽  
Vol 147 (5) ◽  
pp. 1337-1352 ◽  
Author(s):  
Joël Bellaïche ◽  
Gaëtan Chenevier
Keyword(s):  
Number Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Automorphic Representation ◽  
Unitary Groups ◽  
Complex Conjugate ◽  
Automorphic Representations ◽  
Group A ◽  
Cuspidal Automorphic Representations ◽  
Totally Real

AbstractLet K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

scholarly journals A local-global question in automorphic forms

2013 ◽  
Vol 149 (6) ◽  
pp. 959-995 ◽  
Author(s):  
U. K. Anandavardhanan ◽  
Dipendra Prasad
Keyword(s):  
Number Field ◽  
Automorphic Forms ◽  
Quadratic Extension ◽  
Automorphic Representation ◽  
Automorphic Representations ◽  
Formula One ◽  
Distinguished Representations ◽  
Analyze Period ◽  
Cuspidal Automorphic Representations ◽  
Periods Of Automorphic Forms

AbstractIn this paper, we consider the $\mathrm{SL} (2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\mathrm{GL} (2)$: one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$ of cuspidal automorphic representations on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{E} )$ where $E$ is a quadratic extension of a number field $F$, and the other due to Waldspurger involving toric periods of automorphic forms on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$. In both these cases, now involving $\mathrm{SL} (2)$, we analyze period integrals on global$L$-packets; we prove that under certain conditions, a global automorphic $L$-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.

On the generalized Euler–Stieltjes constants for the Rankin–Selberg L-function

2016 ◽  
Vol 13 (06) ◽  
pp. 1363-1379
Author(s):  
Almasa Odžak ◽  
Lejla Smajlović
Keyword(s):  
Asymptotic Formula ◽  
Number Field ◽  
Taylor Series ◽  
Logarithmic Derivative ◽  
Taylor Series Expansion ◽  
Finite Degree ◽  
Automorphic Representations ◽  
Stieltjes Constants ◽  
Orthogonality Relations ◽  
Cuspidal Automorphic Representations

Let [Formula: see text] be a number field of a finite degree and let [Formula: see text] be the Rankin–Selberg [Formula: see text]-function associated to unitary cuspidal automorphic representations [Formula: see text] and [Formula: see text] of [Formula: see text] and [Formula: see text], respectively. The main result of the paper is an asymptotic formula for evaluation of coefficients appearing in the Laurent (Taylor) series expansion of the logarithmic derivative of the function [Formula: see text] at [Formula: see text]. As a corollary, we derive orthogonality and weighted orthogonality relations.

Strong multiplicity one for Siegel cusp forms of degree two

2021 ◽  
Vol 33 (5) ◽  
pp. 1157-1167
Author(s):  
Arvind Kumar ◽  
Jaban Meher ◽  
Karam Deo Shankhadhar
Keyword(s):  
Symplectic Group ◽  
Cusp Forms ◽  
Multiplicity One ◽  
Strong Multiplicity

Abstract We prove strong multiplicity one results for Siegel eigenforms of degree two for the symplectic group Sp 4 ⁡ ( ℤ ) {\operatorname{Sp}_{4}(\mathbb{Z})} .

scholarly journals Lower bounds for Maass forms on semisimple groups

2020 ◽  
Vol 156 (5) ◽  
pp. 959-1003
Author(s):  
Farrell Brumley ◽  
Simon Marshall
Keyword(s):  
Real Number ◽  
Number Field ◽  
Lower Bounds ◽  
Positive Curvature ◽  
Semisimple Group ◽  
Maass Forms ◽  
Semisimple Groups ◽  
Cartan Involution ◽  
Totally Real ◽  
Real Number Field

Let $G$ be an anisotropic semisimple group over a totally real number field $F$. Suppose that $G$ is compact at all but one infinite place $v_{0}$. In addition, suppose that $G_{v_{0}}$ is $\mathbb{R}$-almost simple, not split, and has a Cartan involution defined over $F$. If $Y$ is a congruence arithmetic manifold of non-positive curvature associated with $G$, we prove that there exists a sequence of Laplace eigenfunctions on $Y$ whose sup norms grow like a power of the eigenvalue.

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